Modifications for the Differential Evolution Algorithm
Abstract
:1. Introduction
2. Modifications
2.1. The Base Algorithm
Algorithm 1: DE algorithm. 

2.2. The New Termination Rule
2.3. The New Differential Weight
3. Experiments
3.1. Test Functions
 Bf1 (Bohachevsky 1) function defined as:$$f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}\frac{3}{10}cos\left(3\pi {x}_{1}\right)\frac{4}{10}cos\left(4\pi {x}_{2}\right)+\frac{7}{10}$$
 Bf2 (Bohachevsky 2) function defined as:$$f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}\frac{3}{10}cos\left(3\pi {x}_{1}\right)cos\left(4\pi {x}_{2}\right)+\frac{3}{10}$$
 Branin function. The function is defined by $f\left(x\right)={\left({x}_{2}\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}6\right)}^{2}+10\left(1\frac{1}{8\pi}\right)cos\left({x}_{1}\right)+10$ with $5\le {x}_{1}\le 10,\phantom{\rule{4pt}{0ex}}0\le {x}_{2}\le 15$. The value of global minimum is 0.397887.with $x\in {[10,10]}^{2}$. The value of global minimum is −0.352386.
 CM function. The Cosine Mixture function is given by the equation:$$f\left(x\right)=\sum _{i=1}^{n}{x}_{i}^{2}\frac{1}{10}\sum _{i=1}^{n}cos\left(5\pi {x}_{i}\right)$$
 Camel function. The function is given by:$$f\left(x\right)=4{x}_{1}^{2}2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}4{x}_{2}^{2}+4{x}_{2}^{4},\phantom{\rule{1.em}{0ex}}x\in {[5,5]}^{2}$$
 Easom function. The function is given by the equation:$$f\left(x\right)=cos\left({x}_{1}\right)cos\left({x}_{2}\right)exp\left({\left({x}_{2}\pi \right)}^{2}{\left({x}_{1}\pi \right)}^{2}\right)$$
 Exponential function, defined as:$$f\left(x\right)=exp\left(0.5\sum _{i=1}^{n}{x}_{i}^{2}\right),\phantom{\rule{1.em}{0ex}}1\le {x}_{i}\le 1$$The global minimum is located at ${x}^{*}=(0,0,...,0)$ with value $1$. In our experiments we used this function with $n=2,4,8,16,32$.
 Goldstein and Price functionThe function is given by the equation:$$\begin{array}{ccc}\hfill f\left(x\right)& =& \left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\right.\hfill \\ & & \left(1914{x}_{1}+3{x}_{1}^{2}14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)]\times \hfill \\ & & [30+{\left(2{x}_{1}3{x}_{2}\right)}^{2}\hfill \\ & & \left(1832{x}_{1}+12{x}_{1}^{2}+48{x}_{2}36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)]\hfill \end{array}$$
 Griewank2 function. The function is given by:$$f\left(x\right)=1+\frac{1}{200}\sum _{i=1}^{2}{x}_{i}^{2}\prod _{i=1}^{2}\frac{cos\left({x}_{i}\right)}{\sqrt{\left(i\right)}},\phantom{\rule{1.em}{0ex}}x\in {[100,100]}^{2}$$The global minimum is located at the ${x}^{*}=(0,0,...,0)$ with value 0.
 Gkls function. $f\left(x\right)=\mathrm{Gkls}(x,n,w)$, is a function with w local minima, described in [71] with $x\in {[1,1]}^{n}$ and n a positive integer between 2 and 100. The value of the global minimum is −1 and in our experiments we have used $n=2,3$ and $w=50,\phantom{\rule{4pt}{0ex}}100$.
 Hansen function. $f\left(x\right)={\sum}_{i=1}^{5}icos\left[(i1){x}_{1}+i\right]{\sum}_{j=1}^{5}jcos\left[(j+1){x}_{2}+j\right]$, $x\in $${[10,10]}^{2}$.
 Hartman 3 function. The function is given by:$$f\left(x\right)=\sum _{i=1}^{4}{c}_{i}exp\left(\sum _{j=1}^{3}{a}_{ij}{\left({x}_{j}{p}_{ij}\right)}^{2}\right)$$$$p=\left(\begin{array}{ccc}0.3689& 0.117& 0.2673\\ 0.4699& 0.4387& 0.747\\ 0.1091& 0.8732& 0.5547\\ 0.03815& 0.5743& 0.8828\end{array}\right)$$
 Hartman 6 function.$$f\left(x\right)=\sum _{i=1}^{4}{c}_{i}exp\left(\sum _{j=1}^{6}{a}_{ij}{\left({x}_{j}{p}_{ij}\right)}^{2}\right)$$$$p=\left(\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ 0.2329& 0.4135& 0.8307& 0.3736& 0.1004& 0.9991\\ 0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ 0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right)$$
 Potential function. The molecular conformation corresponding to the global minimum of the energy of N atoms interacting via the LennardJones potential [72] is used as a test case here. The function to be minimized is given by:$${V}_{LJ}\left(r\right)=4\u03f5\left[{\left(\frac{\sigma}{r}\right)}^{12}{\left(\frac{\sigma}{r}\right)}^{6}\right]$$In the current experiments three different cases were studied: $N=3,\phantom{\rule{4pt}{0ex}}4,\phantom{\rule{4pt}{0ex}}5.$
 Rastrigin function. The function is given by:$$f\left(x\right)={x}_{1}^{2}+{x}_{2}^{2}cos\left(18{x}_{1}\right)cos\left(18{x}_{2}\right),\phantom{\rule{1.em}{0ex}}x\in {[1,1]}^{2}$$The global minimum is located at ${x}^{*}=(0,0)$ with value −2.0.
 Rosenbrock function.This function is given by:$$f\left(x\right)=\sum _{i=1}^{n1}\left(100{\left({x}_{i+1}{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}1\right)}^{2}\right),\phantom{\rule{1.em}{0ex}}30\le {x}_{i}\le 30.$$The global minimum is located at the ${x}^{*}=(0,0,...,0)$ with $f\left({x}^{*}\right)=0$. In our experiments we used this function with $n=4,\phantom{\rule{4pt}{0ex}}8,\phantom{\rule{4pt}{0ex}}16$.
 Shekel 7 function.
 Shekel 5 function.
 Shekel 10 function.
 Sinusoidal function. The function is given by:$$f\left(x\right)=\left(2.5\prod _{i=1}^{n}sin\left({x}_{i}z\right)+\prod _{i=1}^{n}sin\left(5\left({x}_{i}z\right)\right)\right),\phantom{\rule{1.em}{0ex}}0\le {x}_{i}\le \pi .$$The global minimum is located at ${x}^{*}=(2.09435,2.09435,...,2.09435)$ with $f\left({x}^{*}\right)=3.5$. In our experiments we used $n=4,8,16,32$ and $z=\frac{\pi}{6}$ and the corresponding functions are denoted by the labels SINU4, SINU8, SINU16 and SINU32, respectively.
 Test2N function. This function is given by the equation:$$f\left(x\right)=\frac{1}{2}\sum _{i=1}^{n}{x}_{i}^{4}16{x}_{i}^{2}+5{x}_{i},\phantom{\rule{1.em}{0ex}}{x}_{i}\in [5,5].$$The function has ${2}^{n}$ in the specified range and in our experiments we used $n=4,5,6,7$. The corresponding values of global minimum is −156.664663 for $n=4$, −195.830829 for $n=5$, −234.996994 for $n=6$ and −274.163160 for $n=7$.
 Test30N function. This function is given by:$$f\left(x\right)=\frac{1}{10}{sin}^{2}\left(3\pi {x}_{1}\right)\sum _{i=2}^{n1}\left({\left({x}_{i}1\right)}^{2}\left(1+{sin}^{2}\left(3\pi {x}_{i+1}\right)\right)\right)+{\left({x}_{n}1\right)}^{2}\left(1+{sin}^{2}\left(2\pi {x}_{n}\right)\right)$$
3.2. Experimental Results
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
 Kudyshev, Z.A.; Kildishev, A.V.; Boltasseva, V.M.S.A. Machine learning–assisted global optimization of photonic devices. Nanophotonics 2021, 10, 371–383. [Google Scholar] [CrossRef]
 Ding, X.L.; Li, Z.Y.; Meng, J.H.; Zhao, Y.X.; Sheng, G.H. Densityfunctional global optimization of (LA_{2}O_{3})_{n} Clusters. J. Chem. Phys. 2012, 137, 214311. [Google Scholar] [CrossRef] [PubMed]
 Morita, S.; Naoki, N. Global optimization of tensor renormalization group using the corner transfer matrix. Phys. Rev. B 2021, 103, 045131. [Google Scholar] [CrossRef]
 Heiles, S.; Johnston, R.L. Global optimization of clusters using electronic structure methods. Int. J. Quantum Chem. 2013, 113, 2091–2109. [Google Scholar] [CrossRef]
 Yang, Y.; Pan, T.; Zhang, J. Global Optimization of Norris Derivative Filtering with Application for NearInfrared Analysis of Serum Urea Nitrogen. Am. J. Anal. Chem. 2019, 10, 143–152. [Google Scholar] [CrossRef] [Green Version]
 Grebner, C.; Becker, J.; Weber, D.; Engels, B. Tabu search based global optimization algorithms for problems in computational Chemistry. J. Cheminf. 2012, 4, 10. [Google Scholar] [CrossRef]
 Dittner, M.; Müller, J.; Aktulga, H.M.; Hartke, B.J. Efficient global optimization of reactive forcefield parameters. Comput. Chem. 2015, 36, 1550–1561. [Google Scholar] [CrossRef]
 Zhao, W.; Wang, L.; Zhang, Z. SupplyDemandBased Optimization: A Novel EconomicsInspired Algorithm for Global Optimization. IEEE Access 2019, 7, 73182–73206. [Google Scholar] [CrossRef]
 Mishra, S.K. Global Optimization of Some Difficult Benchmark Functions by HostParasite CoEvolutionary Algorithm. Econ. Bull. 2013, 33, 1–18. [Google Scholar]
 Freisleben, B.; Merz, P. A genetic local search algorithm for solving symmetric and asymmetric traveling salesman problems. In Proceedings of the IEEE International Conference on Evolutionary Computation, Nagoya, Japan, 20–22 May 1996; pp. 616–621. [Google Scholar]
 Grbić, R.; Nyarko, E.K.; Scitovski, R. A modification of the DIRECT method for Lipschitz global optimization for a symmetric function. J. Glob. Optim. 2013, 57, 1193–1212. [Google Scholar] [CrossRef]
 Scitovski, R. A new global optimization method for a symmetric Lipschitz continuous function and the application to searching for a globally optimal partition of a onedimensional set. J. Glob. Optim. 2017, 68, 713–727. [Google Scholar] [CrossRef]
 Kim, Y. An unconstrained global optimization framework for real symmetric eigenvalue problems. Appl. Num. Math. 2019, 144, 253–275. [Google Scholar] [CrossRef]
 Osaba, E.; Yang, X.S.; Diaz, F.; LopezGarcia, P.; Carballedo, R. An improved discrete bat algorithm for symmetric and asymmetric Traveling Salesman Problems. Eng. Appl. Artif. Intell. 2016, 49, 59–71. [Google Scholar] [CrossRef]
 Bremermann, H.A. A method for unconstrained global optimization. Math. Biosci. 1970, 9, 1–15. [Google Scholar] [CrossRef]
 Jarvis, R.A. Adaptive global search by the process of competitive evolution. IEEE Trans. Syst. Man Cybergen. 1975, 75, 297–311. [Google Scholar] [CrossRef]
 Price, W.L. Global Optimization by Controlled Random Search. Comput. J. 1977, 20, 367–370. [Google Scholar] [CrossRef] [Green Version]
 Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by simulated annealing. Science 1983, 220, 671–680. [Google Scholar] [CrossRef]
 Van Laarhoven, P.J.M.; Aarts, E.H.L. Simulated Annealing: Theory and Applications; Riedel, D., Ed.; Springer: Dordrecht, The Netherlands, 1987. [Google Scholar]
 Goffe, W.L.; Ferrier, G.D.; Rogers, J. Global Optimization of Statistical Functions with Simulated Annealing. J. Econom. 1994, 60, 65–100. [Google Scholar] [CrossRef] [Green Version]
 Goldberg, D. Genetic Algorithms in Search, Optimization and Machine Learning; AddisonWesley Publishing Company: Reading, MA, USA, 1989. [Google Scholar]
 Michaelewicz, Z. Genetic Algorithms + Data Structures = Evolution Programs; Springer: Berlin, Germany, 1996. [Google Scholar]
 Akay, B.; Karaboga, D. A modified Artificial Bee Colony algorithm for realparameter optimization. Inf. Sci. 2012, 192, 120–142. [Google Scholar] [CrossRef]
 Zhu, G.; Kwong, S. Gbestguided artificial bee colony algorithm for numerical function optimization. Appl. Math. Comput. 2010, 217, 3166–3173. [Google Scholar] [CrossRef]
 Dorigo, M.; Birattari, M.; Stutzle, T. Ant colony optimization. IEEE Comput. Intell. Mag. 2006, 1, 28–39. [Google Scholar] [CrossRef]
 Kennedy, J.; Everhart, R.C. Particle Swarm Optimization. In Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, 27 November–1 December 1995; IEEE Press: Piscataway, NJ, USA, 1995; Volume 4, pp. 1942–1948. [Google Scholar]
 Storn, R. On the usage of differential evolution for function optimization. In Proceedings of the North American Fuzzy Information Processing, Berkeley, CA, USA, 19–22 June 1996; pp. 519–523. [Google Scholar]
 Zhou, Y.; Tan, Y. GPUbased parallel particle swarm optimization. In Proceedings of the 2009 IEEE Congress on Evolutionary Computation, Trondheim, Norway, 18–21 May 2009; pp. 1493–1500. [Google Scholar]
 Dawson, L.; Stewart, I. Improving Ant Colony Optimization performance on the GPU using CUDA. In Proceedings of the 2013 IEEE Congress on Evolutionary Computation, Cancun, Mexico, 20–23 June 2013; pp. 1901–1908. [Google Scholar]
 Barkalov, K.; Gergel, V. Parallel global optimization on GPU. J. Glob. Optim. 2016, 66, 3–20. [Google Scholar] [CrossRef]
 Li, Y.H.; Wang, J.Q.; Wang, X.J.; Zhao, Y.L.; Lu, X.H.; Liu, D.L. Community Detection Based on Differential Evolution Using Social Spider Optimization. Symmetry 2017, 9, 183. [Google Scholar] [CrossRef] [Green Version]
 Yang, W.; Siriwardane, E.M.D.; Dong, R.; Li, Y.; Hu, J. Crystal structure prediction of materials with high symmetry using differential evolution. J. Phys. Condens. Matter 2021, 33, 455902. [Google Scholar] [CrossRef] [PubMed]
 Lee, C.Y.; Hung, C.H. Feature Ranking and Differential Evolution for Feature Selection in Brushless DC Motor Fault Diagnosis. Symmetry 2021, 13, 1291. [Google Scholar] [CrossRef]
 Saha, S.; Das, R. Exploring differential evolution and particle swarm optimization to develop some symmetrybased automatic clustering techniques: Application to gene clustering. Neural Comput. Appl. 2018, 30, 735–757. [Google Scholar] [CrossRef]
 Wu, Z.; Cui, N.; Zhao, L.; Han, L.; Hu, X.; Cai, H.; Gong, D.; Xing, L.; Chen, X.; Zhu, B.; et al. Estimation of maize evapotranspiration in semihumid regions of Northern China Using PenmanMonteith model and segmentally optimized Jarvis model. J. Hydrol. 2022, 22, 127483. [Google Scholar] [CrossRef]
 TleloCuautle, E.; GonzlezZapata, A.M.; DazMuoz, J.D.; Fraga, L.G.D.; CruzVega, I. Optimization of fractionalorder chaotic cellular neural networks by metaheuristics. Eur. Phys. J. Spec. Top. 2022. Available online: https://link.springer.com/article/10.1140/epjs/s11734022004526 (accessed on 25 January 2022).
 Sun, G.; Li, C.; Deng, L. An adaptive regeneration framework based on search space adjustment for differential evolution. Neural Comput. Appl. 2021, 33, 9503–9519. [Google Scholar] [CrossRef]
 Civiciogluan, P.; Besdok, E. Bernstainsearch differential evolution algorithm for numerical function optimization. Expert Syst. Appl. 2019, 138, 112831. [Google Scholar] [CrossRef]
 Liang, J.; Qiao, K.; Yu, K.; Ge, S.; Qu, B.; Li, R.X.K. Parameters estimation of solar photovoltaic models via a selfadaptive ensemblebased differential evolution. Solar Energy 2020, 207, 336–346. [Google Scholar] [CrossRef]
 Peng, L.; Liu, S.; Liu, R.; Wang, L. Effective long shortterm memory with differential evolution algorithm for electricity price prediction. Energy 2018, 162, 1301–1314. [Google Scholar] [CrossRef]
 Awad, N.; Hutter, N.M.A.F. Differential Evolution for Neural Architecture Search. In Proceedings of the 1st Workshop on Neural Architecture Search, Addis Ababa, Ethiopia, 26 April 2020. [Google Scholar]
 Ilonen, J.; Kamarainen, J.K.; Lampinen, J. Differential Evolution Training Algorithm for FeedForward Neural Networks. Neural Process. Lett. 2003, 17, 93–105. [Google Scholar] [CrossRef]
 Slowik, A. Application of an Adaptive Differential Evolution Algorithm With Multiple Trial Vectors to Artificial Neural Network Training. IEEE Trans. Ind. Electron. 2011, 58, 3160–3167. [Google Scholar] [CrossRef]
 Wang, L.; Zeng, Y.; Chen, T. Back propagation neural network with adaptive differential evolution algorithm for time series forecasting. Expert Syst. Appl. 2015, 42, 855–863. [Google Scholar] [CrossRef]
 Wang, X.; Xu, G. Hybrid Differential Evolution Algorithm for Traveling Salesman Problem. Procedia Eng. 2011, 15, 2716–2720. [Google Scholar] [CrossRef] [Green Version]
 Ali, I.M.; Essam, D.; Kasmarik, K. A novel design of differential evolution for solving discrete traveling salesman problems. Swarm Evolut. Comput. 2020, 52, 100607. [Google Scholar] [CrossRef]
 Liu, J.; Lampinen, J. A differential evolution based incremental training method for RBF networks. In Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation (GECCO ’05), Washington, DC, USA, 25–29 June 2005; pp. 881–888. [Google Scholar]
 O’Hora, B.; Perera, J.; Brabazon, A. Designing Radial Basis Function Networks for Classification Using Differential Evolution. In Proceedings of the 2006 IEEE International Joint Conference on Neural Network Proceedings, Vancouver, BC, Canada, 16–21 July 2006; pp. 2932–2937. [Google Scholar]
 Naveen, N.; Ravi, V.; Rao, C.R.; Chauhan, N. Differential evolution trained radial basis function network: Application to bankruptcy prediction in banks. Int. J. BioInspir. Comput. 2010, 2, 222–232. [Google Scholar] [CrossRef]
 Chen, Z.; Jiang, X.; Li, J.; Li, S.; Wang, L. PDECO: Parallel differential evolution for clusters optimization. J. Comput. Chem. 2013, 34, 1046–1059. [Google Scholar] [CrossRef]
 Ghosh, A.; Mallipeddi, R.; Das, S.; Das, A. A Switched Parameter Differential Evolution with Multidonor Mutation and Annealing Based Local Search for Optimization of LennardJones Atomic Clusters. In Proceedings of the 2018 IEEE Congress on Evolutionary Computation (CEC), Rio de Janeiro, Brazil, 8–13 July 2018; pp. 1–8. [Google Scholar]
 Zhang, Y.; Zhang, H.; Cai, J.; Yang, B. A Weighted Voting Classifier Based on Differential Evolution. Abstr. Appl. Anal. 2014, 2014, 376950. [Google Scholar] [CrossRef]
 Maulik, U.; Saha, I. Automatic Fuzzy Clustering Using Modified Differential Evolution for Image Classification. IEEE Trans. Geosci. Remote Sens. 2010, 48, 3503–3510. [Google Scholar] [CrossRef]
 Hancer, E. Differential evolution for feature selection: A fuzzy wrapper–filter approach. Soft Comput. 2019, 23, 5233–5248. [Google Scholar] [CrossRef]
 Vivekanandan, T.; Iyengar, N.C.S.N. Optimal feature selection using a modified differential evolution algorithm and its effectiveness for prediction of heart disease. Comput. Biol. Med. 2017, 90, 125–136. [Google Scholar] [CrossRef] [PubMed]
 Deng, W.; Liu, H.; Xu, J.; Zhao, H.; Song, Y. An Improved QuantumInspired Differential Evolution Algorithm for Deep Belief Network. IEEE Trans. Instrum. Meas. 2020, 69, 7319–7327. [Google Scholar] [CrossRef]
 Wu, T.; Li, X.; Zhou, D.; Li, N.; Shi, J. Differential Evolution Based LayerWise Weight Pruning for Compressing Deep Neural Networks. Sensors 2021, 21, 880. [Google Scholar] [CrossRef] [PubMed]
 Mininno, E.; Neri, F.; Cupertino, F.; Naso, D. Compact Differential Evolution. IEEE Trans. Evolut. Comput. 2011, 15, 32–54. [Google Scholar] [CrossRef]
 Qin, A.K.; Huang, V.L.; Suganthan, P.N. Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization. IEEE Trans. Evolut. Comput. 2009, 13, 398–417. [Google Scholar] [CrossRef]
 Liu, J.; Lampinen, J. A Fuzzy Adaptive Differential Evolution Algorithm. Soft Comput. 2005, 9, 448–462. [Google Scholar] [CrossRef]
 Wang, H.; Rahnamayan, S.; Wu, Z. Parallel differential evolution with selfadapting control parameters and generalized oppositionbased learning for solving highdimensional optimization problems. J. Parallel Distrib. Comput. 2013, 73, 62–73. [Google Scholar] [CrossRef]
 Das, S.; Mullick, S.S.; Suganthan, P.N. Recent advances in differential evolution—An updated survey. Swarm Evolut. Comput. 2016, 27, 1–30. [Google Scholar] [CrossRef]
 Ali, M.M.; Törn, A. Population setbased global optimization algorithms: Some modifications and numerical studies. Comput. Oper. Res. 2004, 31, 1703–1725. [Google Scholar] [CrossRef] [Green Version]
 Ali, M.M. Charoenchai Khompatraporn, Zelda B. Zabinsky, A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems. J. Glob. Opt. 2005, 31, 635–672. [Google Scholar] [CrossRef]
 Floudas, C.A.; Pardalos, P.M.; Adjiman, C.; Esposoto, W.; Gümüs, Z.; Harding, S.; Klepeis, J.; Meyer, C.; Schweiger, C. Handbook of Test Problems in Local and Global Optimization; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999. [Google Scholar]
 Ali, M.M.; Kaelo, P. Improved particle swarm algorithms for global optimization. Appl. Math. Comput. 2008, 196, 578–593. [Google Scholar] [CrossRef]
 Koyuncu, H.; Ceylan, R. A PSO based approach: Scout particle swarm algorithm for continuous global optimization problems. J. Comput. Des. Eng. 2019, 6, 129–142. [Google Scholar] [CrossRef]
 Siarry, P.; Berthiau, G.; Durdin, F.F.; Haussy, J. Enhanced simulated annealing for globally minimizing functions of manycontinuous variables. ACM Trans. Math. Softw. 1997, 23, 209–228. [Google Scholar] [CrossRef]
 Tsoulos, I.G.; Lagaris, I.E. GenMin: An enhanced genetic algorithm for global optimization. Comput. Phys. Commun. 2008, 178, 843–851. [Google Scholar] [CrossRef]
 Powell, M.J.D. A Tolerant Algorithm for Linearly Constrained Optimization Calculations. Math. Programm. 1989, 45, 547–566. [Google Scholar] [CrossRef]
 Gaviano, M.; Ksasov, D.E.; Lera, D.; Sergeyev, Y.D. Software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 2003, 29, 469–480. [Google Scholar] [CrossRef]
 LennardJones, J.E. On the Determination of Molecular Fields. Proc. R. Soc. Lond. A 1924, 106, 463–477. [Google Scholar]
Parameter  Value 

NP  10n 
F  0.8 
CR  0.9 
M  20 
$\u03f5$  ${10}^{4}$ 
Function  Static  Ali  Proposed 

BF1  1142  1431  847 
BF2  1164  1379  896 
BRANIN  984  816  707 
CM4  3590  7572  2079 
CAMEL  1094  18,849  685 
EASOM  1707  2014  1327 
EXP2  532  323  449 
EXP4  2421  1019  1494 
EXP8  15,750  3670  5632 
EXP16  160,031  15,150  21,416 
EXP32  320,039  152,548  77,936 
GKLS250  784  944  614 
GKLS2100  772  1531  599 (0.97) 
GKLS350  1906 (0.93)  3263  1275 (0.93) 
GKLS3100  1883  3539  1373 
GOLDSTEIN  988  818  769 
GRIEWANK2  1299 (0.97)  1403  883 (0.93) 
HANSEN  2398  2968  1400 
HARTMAN3  1448  836  1050 
HARTMAN6  9489(0.97)  4015(0.97)  4667(0.80) 
POTENTIAL3  90,027  89,776  21,824 
POTENTIAL4  120,387 (0.97)  120,405 (0.33)  45,705 (0.97) 
POTENTIAL5  150,073  150,104  83,342 
RASTRIGIN  1246  1098 (0.93)  871 
ROSENBROCK4  6564  9695  4499 
ROSENBROCK8  44,240  72,228  13,959 
ROSENBCROK16  160,349 (0.90)  160,538 (0.60)  53,594 
SHEKEL5  5524  3810  3057 (0.83) 
SHEKEL7  5266  3558  2992 (0.87) 
SHEKEL10  5319  3379  3076 
TEST2N4  4200  1980  2592 
TEST2N5  7357  2957  4055 
TEST2N6  12,074  4159  5836 
TEST2N7  18,872  5490  7904 
SINU4  3270  1855  2216 
SINU8  23,108  6995  8135 
SINU16  160,092  36,044  30,943 
SINU32  213,757 (0.70)  160,536 (0.53)  83,369 (0.80) 
TEST30N3  1452  1732  959 
TEST30N4  1917  2287  1378 
Total  1,564,515 (0.97)  1,062,714 (0.96)  506,404 (0.98) 
Function  Static  Ali  Proposed 

BF1  996  1124  889 
BF2  926  1026  816 
BRANIN  878  900  730 
CM4  1148 (0.70)  1991  1103 
CAMEL  1049  904 (0.93)  846 
EASOM  447  448  446 
EXP2  470  461  467 
EXP4  915  903  892 
EXP8  1797  3558  1796 
EXP16  3578  7082  3521 
EXP32  7082  14,125  7022 
GKLS250  498  576  493 
GKLS2100  533  884 (0.97)  515 
GKLS350  823  1130 (0.93)  814 (0.97) 
GKLS3100  858  1495 (0.97)  829 (0.93) 
GOLDSTEIN  945  993  915 
GRIEWANK2  947  921  826 
HANSEN  2104  1949  1479 
HARTMAN3  1017  1005  952 
HARTMAN6  4679 (0.90)  3744 (0.97)  3128 (0.87) 
POTENTIAL3  21,473  2284  8197 
POTENTIAL4  44,191 (0.43)  3098 (0.33)  24,659 (0.97) 
POTENTIAL5  75,910  3443  52,664 
RASTRIGIN  841  994  777 
ROSENBROCK4  4934  7192  3300 
ROSENBROCK8  29,583  49,696  10,907 
ROSENBCROK16  160,349  160,538 (0.60)  38,315 
SHEKEL5  4389 (0.97)  4266  2839 (0.83) 
SHEKEL7  3905  3685  2668 
SHEKEL10  4049  3548  2629 
TEST2N4  2785  2275  2221 
TEST2N5  4481  3170  3122 
TEST2N6  6852  4286  4296 
TEST2N7  11971  5701  6267 
SINU4  2322  1987  1755 
SINU8  9990  6156  5113 
SINU16  6892  3628 (0.97)  16,905 
SINU32  7235 (0.80)  7438 (0.83)  7218 
TEST30N3  1033  1098  951 
TEST30N4  1355  1444  1285 
Total  432,610 (0.98)  321,166 (0.96)  224,567 (0.99) 
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. 
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Charilogis, V.; Tsoulos, I.G.; Tzallas, A.; Karvounis, E. Modifications for the Differential Evolution Algorithm. Symmetry 2022, 14, 447. https://doi.org/10.3390/sym14030447
Charilogis V, Tsoulos IG, Tzallas A, Karvounis E. Modifications for the Differential Evolution Algorithm. Symmetry. 2022; 14(3):447. https://doi.org/10.3390/sym14030447
Chicago/Turabian StyleCharilogis, Vasileios, Ioannis G. Tsoulos, Alexandros Tzallas, and Evangelos Karvounis. 2022. "Modifications for the Differential Evolution Algorithm" Symmetry 14, no. 3: 447. https://doi.org/10.3390/sym14030447